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Traces and Determinants of Pseudodifferential Operators$
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Simon Scott

Print publication date: 2010

Print ISBN-13: 9780198568360

Published to Oxford Scholarship Online: January 2011

DOI: 10.1093/acprof:oso/9780198568360.001.0001

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Geometric Families of Pseudodifferential Operators and Determinant Line Bundles

Geometric Families of Pseudodifferential Operators and Determinant Line Bundles

Chapter:
(p.598) V Geometric Families of Pseudodifferential Operators and Determinant Line Bundles
Source:
Traces and Determinants of Pseudodifferential Operators
Author(s):

Simon Scott

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198568360.003.0006

The final chapter focuses on applications of trace and determinant structures to geometric families of pseudodifferential operators. This is closely entwined with developments in QFT and string theory. The first section reviews the construction of families of pseudodifferential operators parameterized by a fibration. A generalized trace on complex powers of geometric families is used to construct zeta trace forms. Specifically,a natural logarithm map from geometric families to the de Rham algebra on the parameter manifold is constructed whose superconnection character form is a canonical representative for the Chern class. In cohomology this is a canonical form representative for the index bundle. The final part of the chapter gives a of the determinant line bundle endowed with its zeta function metric,. The chapter closes with a detailed computation of the zeta metric for the case of a family of Cauchy Riemann operators on a surface.

Keywords:   Families of pseudodifferential operators, fibrations, local families index theorem, determinant line bundle, Quillen metric, superconnection

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